Abstract: The alternating series is , with f a single-signed monotonic function of the real variable x. The are , their sign fixed by repetition of the 'template' [j] of finite length 2p. [j] constitutes a difference scheme of 'differential order' D, which can be determined. The principal theorem is that is 'partially convergent' if and only if is bounded. A series is partially convergent when the limit as of the sum of 2pM terms exists. For [j] 'pure', the improved Euler-Maclaurin expansion (IEM) gives the compact representation (A)

is the sum, is the Dth 'template moment', and the are Bernoulli numbers. Efficient means for practical summation of these series follow also from IEM. In illustration, 10 alternating series with D ranging from 1 to 3 are summed using IEM. It is found that the leading term of (A) with gives a simple but effective estimate of sums. The paper also gives a comparison with Euler's transformation in the case and discusses sums to N terms with nonintegral and finite but large.